Question: Let $ABCD$ be a convex quadrilateral, and let $G_A,$ $G_B,$ $G_C,$ $G_D$ denote the centroids of triangles $BCD,$ $ACD,$ $ABD,$ and $ABC,$ respectively.  Find $\frac{[G_A G_B G_C G_D]}{[ABCD]}.$

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unitsize(0.6 cm);

pair A, B, C, D;
pair[] G;

A = (0,0);
B = (7,1);
C = (5,-5);
D = (1,-3);
G[1] = (B + C + D)/3;
G[2] = (A + C + D)/3;
G[3] = (A + B + D)/3;
G[4] = (A + B + C)/3;

draw(A--B--C--D--cycle);
draw(G[1]--G[2]--G[3]--G[4]--cycle,red);

label("$A$", A, W);
label("$B$", B, NE);
label("$C$", C, SE);
label("$D$", D, SW);
dot("$G_A$", G[1], SE);
dot("$G_B$", G[2], W);
dot("$G_C$", G[3], NW);
dot("$G_D$", G[4], NE);
[/asy]
Explanation: We have that
\begin{align*}
\overrightarrow{G}_A &= \frac{\overrightarrow{B} + \overrightarrow{C} + \overrightarrow{D}}{3}, \\
\overrightarrow{G}_B &= \frac{\overrightarrow{A} + \overrightarrow{C} + \overrightarrow{D}}{3}, \\
\overrightarrow{G}_C &= \frac{\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{D}}{3}, \\
\overrightarrow{G}_D &= \frac{\overrightarrow{A} + \overrightarrow{B} + \overrightarrow{C}}{3}.
\end{align*}Then
\begin{align*}
\overrightarrow{G_B G_A} &= \overrightarrow{G_A} - \overrightarrow{G_B} \\
&= \frac{\overrightarrow{B} + \overrightarrow{C} + \overrightarrow{D}}{3} - \frac{\overrightarrow{A} + \overrightarrow{C} + \overrightarrow{D}}{3} \\
&= \frac{1}{3} (\overrightarrow{B} - \overrightarrow{A}) \\
&= \frac{1}{3} \overrightarrow{AB}.
\end{align*}It follows that $\overline{G_B G_A}$ is parallel to $\overline{AB},$ and $\frac{1}{3}$ in length.

Similarly,
\[\overrightarrow{G_B G_C} = \frac{1}{3} \overrightarrow{CB}.\]It follows that $\overline{G_B G_C}$ is parallel to $\overline{BC},$ and $\frac{1}{3}$ in length.  Therefore, triangles $ABC$ and $G_A G_B G_C$ are similar, and
\[[G_A G_B G_C] = \frac{1}{9} [ABC].\]In the same way, we can show that
\[[G_C G_D G_A] = \frac{1}{9} [CDA].\]Therefore, $[G_A G_B G_C G_C] = \frac{1}{9} [ABCD],$ so $\frac{[G_A G_B G_C G_D]}{[ABCD]} = \boxed{\frac{1}{9}}.$